Why the charge is the same for capacitors connected in series? 
For example in this figure, I can understand that the charge on plates 1 and 4, and plate 2 and 3 must have the same value because of charge conservation. Is there any justification that plate 2 must have the same charge as plate 1?
 A: Refer to the posted figure, start with an uncharged capacitor and assume that the free charges in a conductor are positive.  The "H"-shaped piece in the middle (from 2 to 3) has zero net charge.  When the series combination is connected to the battery, it still has zero net charge because there is no path that will allow charge from the outside to flow in it.
However, the conducting piece from "A" to "1" is an equipotential at the potential of "+" terminal of the battery.  The potential was raised by charges amounting to +Q that accumulated on plate "1". Since the entire circuit must have zero net charge, these charges must have come from the conducting piece "4" to "B" whose potential has decreased to match the "-" terminal of the battery.  So you have charge Q removed from "4" and added to "1".
How are the free charges in the "H"-shaped piece in the middle going to respond to that? They will be attracted to "4" and pile up on "3" leaving a deficit of charges at "2".  In other words, the positive charge on plate "3" is the same as the magnitude of the negative charge on plate "2".  Furthermore, these magnitudes must be the same as the magnitudes of the charges on plates "1" and "4" because the entire circuit, as mentioned earlier, has zero net charge.
It's all charge conservation.
A: 
Is there any justification that plate 2 must have the same charge as
plate 1?

In a series circuit the current (charge per unit time) is the same going through all components. That means at any instant in time the positive charge supplied by the positive terminal A making plate 1 positive has to equal the positive charge exiting plate 2 making it equally negative, and so on for all the plates returning to the negative terminal B.
Hope this helps.
